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The F-rational signature and drops in the Hilbert–Kunz multiplicity

### Melvin Hochster and Yongwei Yao

Vol. 16 (2022), No. 8, 1777–1809
##### Abstract

Let $\left(R,\mathfrak{𝔪}\right)$ be a Noetherian local ring of prime characteristic $p$. We define the F-rational signature of $R$, denoted by $r\left(R\right)$, as the infimum, taken over pairs of ideals $I⊊J$ such that $I$ is generated by a system of parameters and $J$ is a strictly larger ideal, of the drops ${e}_{\mathrm{HK}}\left(I,R\right)\right)-{e}_{\mathrm{HK}}\left(J,R\right)$ in the Hilbert–Kunz multiplicity. If $R$ is excellent, then $R$ is F-rational if and only if $r\left(R\right)>0$. The proof of this fact depends on the following result in the sequel: Given an $\mathfrak{𝔪}$-primary ideal $I$ in $R$, there exists a positive ${\delta }_{I}\in {ℝ}^{+}$ such that, for any ideal $J⊋I$, ${e}_{\mathrm{HK}}\left(I,R\right)-{e}_{\mathrm{HK}}\left(J,R\right)$ is either $0$ or at least ${\delta }_{I}$. We study how the F-rational signature behaves under deformation, flat local ring extension, and localization.

##### Keywords
F-rational signature, F-signature, Hilbert–Kunz multiplicity, Frobenius, Cohen–Macaulay, Gorenstein, regular
##### Mathematical Subject Classification 2010
Primary: 13A35
Secondary: 13C13, 13H10